QR 38, 2/13/07Rationality and Expected Utility

I. Rationality

II. Expected utility

III. Sets and probabilities

I. Rationality

• Rationality one of the major assumptions of game theory.

• All players are assumed rational; all know that all the others are rational; etc.

• Definition: rationality means that actors are goal-oriented and calculating.

• That is, each player in a game tries to achieve the highest possible payoff for himself.

Rationality

• More precisely: Each actor has a consistent set of rankings, called values or payoffs. They are consistent in that if outcome A is preferred to outcome B, and B to C, then A is preferred to C (transitive).

• Each player then calculates the strategy that best serves these interests. They assess the value of alternative courses of action and compare them.

What rationality does not mean

• Selfish

• Short-run

• Sharing the same value system as other players or “ethical people”

• No particular content

• No assumption about ranking of outcomes except that they’re consistent

Is rationality a good assumption?

Potential problems:

• What is the unit behaving rationally? State; individual leaders (BdM)?

• The rationality assumption may be harder to accept for corporate or aggregate actors

Is rationality a good assumption?

• Individuals may lack the ability to make the complex calculations required by game theory:– May be “boundedly rational” and use

shortcuts– May make mistakes in calculations– May act on the basis of emotion rather than

calculating expected values

How can we justify the rationality assumption?

• Experimental evidence

• General idea of being motivated by goals and trying to do as well as possible seem reasonable.

• Will be thinking of players as implicitly choosing optimal strategies, even if they do not go through the actual process of calculation that game theory assumes.

II. Expected utilityDefinition: expected utility is the anticipated

payoff to any particular policy choice.• Rational decisionmakers calculate the

expected utility associated with each strategy, and choose the one that gives them the highest expected utility.

• The expected utility (or expected value) of any particular strategy is calculated by considering the probability of each outcome and the value attached to that outcome.

Expected utility example

Columbus game: Begin by considering Spain’s expected utility for each of the options Columbus presented.

• Success = gaining the power associated with Asian trade

• Spain cared about the probability of success.

Calculating expected utility

1. Identify the options available (choices of action)

For Spain, four options: east, west, overland, and doing nothing.

Calculating expected utility

2. Identify the probability associated with each option

Probability of success for each route: pe, pw, po, pn

Ferdinand believed that the probability of success for each route was about the same, very small. The probability of success if he did nothing was zero.

pe=pw=po>pn=0

Calculating expected utility

3. Identify the value of each option. We need to calculate the net benefits: benefits minus costs.

Assume that the gross benefits of doing nothing are equivalent to the benefits associated with the failure of other options, i.e., the status quo prevails.

Identify the benefits of success: be=bw=bo>bn

Calculating expected utility

Costs: the cost of doing nothing is zero, cn=0.

For Ferdinand, the cost of the west route was lower than the cost of the east or overland route: cn<cw<ce, co

Net benefits for each route are b-c

Calculating expected utility

4. Calculate the expected value of each option: what each is expected to yield. This is the expected utility: p(b-c)

General formula:

EU=p1(b1-c1) + p2(b2-c2) + … + px(bx-cx)

The set of all possible outcomes is {1, 2, …, x}

Calculating expected utility

Applying this formula to Ferdinand, for each option:

EUe = pe(be-ce) + (1-pe)(bn-ce)Have to consider the possible outcomes for

each option: here, success and failure. The probability of failure is 1-(probability of

success)

Net benefits of failure are bn-costs of that option (remember above)

Calculating expected utility

EUw = pw(bw-cw) + (1-pw)(bn-cw)

EUo = po(bo-co) + (1-po)(bn-co)

EUn = pn(bn-cn) + (1-pn)(bn-cn)

Comparing expected utilities

To identify the best choice, compare the expected utility of each option; choose the one with the highest expected utility.

cw < ce, co. All other terms are equal.

So, EUw>EUe, EUw>EUo.

Comparing expected utilities

Is EUw>EUn? EUn=0

EUw = pw(bw-cw) + (1-pw)(bn-cw)

= pw(bw-cw) + (1-pw)(-cw)

= pwbw – pwcw – cw + pwcw

= pwbw – cw

Since the benefits of success are very large, and the costs of the west route small, even a small probability of success gives EUw>EUn.

Calculating expected utilities

Could apply same analysis to Portugal (see BdM). Difference is that Portugal’s estimates of probabilities and costs was different because it knew about the eastern route; so this turns out to have the highest EU.

Expected value

D&S use the term “expected value” interchangeably with “expected utility”; could also use the term expected payoffs.

D&S formula (p. 228) same as that in BdM, just different notation.

Think of probabilities as the weights that you put on each outcome.

Expected value

D&S denote the value of the outcome by X (think of X=b-c, to compare to BdM).

The payoff can take n possible values, X1, X2, …, Xn.

The respective probabilities are p1, p2, …, pn.

EU=p1X1+p2X2+…+pnXn

III. Sets and probabilities

To fully understand expected utilities and work with them, you need to know something about sets and probabilities.

Sets

Sets: will come up in a number of contexts as we begin to study game theory.

Sets of:

• outcomes

• strategies

Sets

• Definition: a set is a collection of elements.

• The set of all elements is called the universal set; U.

• If element x belongs to set S, x is a member of S; xS.

• The set containing no elements is called the empty or null set; .

Sets

• If all members of S1 are also members of S2, we say that S1 is a subset of S2 and that S2 contains S1; S2S1.

• Sets are disjoint if they have no members in common.

Sets

• There are 3 basic operations in set theory:– The union of S1 and S2, S1S2, is the set of all

elements that are members of both.– The intersection is all elements that are

members of both; S1S2.

– The complement is the set of all elements that are not members of S1; S1

c.

Examples of manipulating sets

• S1={a, b, c}, S2={d, e, f}, S3={c}

• S1 and S2 are disjoint

• S1S2={a, b, c, d, e, f}

• S1S3={a, b, c}

• S1S2=

• S1S3={c}

• S1c={d, e, f}

Probabilities

• Consider the set of events X in which you are interested; these might be all the possible outcomes of a game.

• Divide this set into some number of subsets Y, Z, …, none of which overlap; which are disjoint.

• The probabilities of each subset occurring must sum to the probability of the full set of events. If that set of events includes all the possible outcomes, its probability is 1.

Probabilities

• A probability is a number between (and including) 0 and 1.

• Put another way: if the occurrence of X requires the occurrence of any one of several disjoint Y, Z, …, then the probability of X is the sum of the separate probabilities of Y, Z, …

Probabilities

• Addition rule: p(X) = p(Y)+p(Z)+…, if Y, Z, … are disjoint (mutually exclusive) and exhaustive.

• p()=0

• Conditional probabilities: the probability of Z given Y is written p(Z|Y).

Probability example

• Consider a game where the possible outcomes are win, lose, or tie (W, L, T)

• P(no T) = p(W) + P(L)

• So, if P(W)=.5, P(L)=.4, then P(no T)=.9

Expectations

• The expectation of X, E(X), is the sum of the possible values of X multiplied by the probability that each occurs

• E(X)=xi(p(xi)), for i=1 to n.

• This is a generalization of the equation for expected utility discussed earlier.

Probabilities

Summary:

• Outcomes are the elements of sets.

• The probability of any outcome or event is between 0 and 1.

• The set of outcomes is exhaustive and mutually exclusive; one (and only one) outcome must occur.